Electrical network



Jan. l2, 1937. c. MzsoN GEWERTZ ELECTR I CAL NETWORK Filed May 5, 1952 ll Sheets-Sheet 2 `Fam. 12, 1937.

c. M=soN GEWERTZ 2,067,444

ELECTRICAL NETWORK Filed May 5, 1952 ll Sheets-Sheet 5 Jan. 12, 1937- c. M;soN GEWERTZ ELECTRICAL NETWORK Filed May 5, 1932 ll Sheets-Sheet 4 @y @ww mf. 6W 5 WM-4 2 Q Hw Jan. 12, 1937. C. MZSON GEWERTZ 2,067,444

ELECTR I CAL NETWORK Jan. 12, 1937.

c. M=soN GEWERTZ 2,067,444

ELECTRICAL NETWORK Fle May 5, 1952 ll Sheets-Sheet 6 Jan. 12, 1937. c. M=soN GEwr-:RTZ

ELECTRICAL NETWORK Filed May 5, 1932 ll Sheets-Sheet 7 `Ian. 12, 1937. Q MON GEWERTZ 2,067,444

ELECTRI CAL NETWORK Jan. 12, 1937. c. M=soN GEwET ELECTRICAL NETWORK Filed May 5, 1932 ll Sheets-Sheet 9 Jan. 12, 1937. c. M;soN GEWERTZ ELECTRICAL NETWORK 11 sheets-sheet io Filed May 5, 1952 y .52m L Jan. 12, 1937. c. M=soN GEWERTZ ELECTRICAL NETWORK Filed May 5, 1932 ll SheetS-Sheetl 11 Patented Jan. 12, 1937 NT @HQE 51 Claims.

The present invention relates to electrical networks and' more particularly to four-terminal networks having certain prescribed characteristics.

The principal object of the present invention is to provide a finite, purely reactive or dissipative, electrical four-terminal network whose two shortcircuit driving-point admittances and short-circuit transfer admittance-or-two open-circuit driving-point impedances and open-circuit transfer impedance are prescribed functions of frequency.

Such a network, which has been wanted for a long period of time, is applicable to the solution of numerous problems arising in the field' of electrical communication engineering and naturally it is also applicable to power engineering, where one single frequency is considered.

It may, for example, be required to join together two systems of communication circuits each having denite admittances (or impedances) by means of a four-terminal network having preassigned driving point admittances (or impedances) and transfer admittance (or impedance) in order to improve the desired electrical properties of the circuit in some way and secure the most desirable operation of the circuit as a whole.

On the other hand, it may be required to change the alternating current admittance (or impedance) of a system, in which case a four-terminal network having preassigned admittances (or impedances) may have to be connected in parallel or series-(or perhaps in chain connection) with it, etc.

Further, it may be desired to design a nite four-terminal network equivalent to a transmission line.

A four-terminal network may be used as a twoterminal network and in such a case the same network oiers four driving-point admitt'ances (impedances) ,-two for short circuit and two for open-circuit at the opposite end.

If the two short-circuit driving-point admittances (impedances), say, are prescribed, then we can readily select the short-circuit transfer admittance such that the necessary and suicient conditions, to be stated in the following, are met with, whereupon we can apply one of my gener-al methods of realization and build the network.

When used as a two-terminal network, a fourterminal network obtained from prescribed driving-point admittance or impedance functions can be used for solving the three general types of problems stated by T, C. Fry in his U. S. Patent 1,570,215, page 1, dealing with a two-terminal electrical network having preassigned drivingpoint impedance-function.

The three types of problems referred to are:

(1) The design of a balancing network employed, for example, in two-way repeater circuits.

(2) The design of a network having a prescribed driving-point impedance such that when two component parts of a circuit are adapted to one another the electrical properties of the circuit as a whole are improved; for instance, the reection losses are decreased.

(3) The design of a network having such properties that it can be used for correcting for the distortion which a signal may experience when transmitted through a given system due to the unavoidable addition of appropriate apparatus at the sending or the receiving end. Here we need a network having such a prescribed impedance that the output current of the distorting system when connected to this impedance will be of the same wave form as the input E. M. F. of the system.

It will be apparent that there are also many other types of problems (for instance, related to relay circuits) to which my networks are applicable, but these need not be discussed here.

The networks according to the present invention have numerous advantages over the networks heretofore available. Practically all known methods in designing corrective and selective networks are based on continuous approximation methods and in order to get a satisfactory result more and more sections have to be added. The network obtained by applying my general methods is, however, in all cases an exact realization of the prescribed functions. It contains, furthermore, a nite number of meshes and elements.

I describe below three general methods of synthesizing four-terminal networks. Method I gives networks containing only two kinds of network elements plus necessary transformers and is claimed in my copending application Serial No. 609,428. Methods II and III on the' other hand give networks which ordinarily contain all three kinds of network elements, i. e., inductance, capacitance and resistance plus necessary transformers. The networks obtained by methods I and III are all passive. With method II negative reactance elements may be introduced. In all cases, however, the network appears to contain the minimum number of elements for the prescribed conditions.

Further, it is only necessary to find the three short-circuit admittances or the three open-circuit impedances from the propagation function, generated from its amplitude frequency characteristic, and two characteristic impedances, whereupon my general methods can always be applied for the design of corrective (amplitude corrective or phase corrective networks or possibly networks which make both corrections simultaneously) and selective (electric lter networks) networks of any kind.

In practice the propagation curve and two characteristic impedances may be prescribed, in which case the first step would be to find an wfunction (where w is=21r times frequency) as a quotient of two integral polynomials fitting this curve and then, as indicated above, this function may generate the propagation function itself, whereupon, lafter that the corresponding three short-circuit admittances or three opencircuit impedances are found, a four-terminal network having such properties that it answers in a prescribed manner to an applied E. M. F. of any frequency can then be designed by the aid of my general methods.

Having the three short-circuit admittances or the three open-circuit impedances given, my general methods of design furnish a possibility to go straight to the point and a compact finite fourterminal network, which embodies the preassigned requirements directly and mathematically exactly, is obtained.

Further details of my invention may loe obtained from the following detailed description and the accompanying drawings, in which Fig. 1 represents a finite passive four-terminal network containing n meshes; Fig. 2 represents a fourterminal network terminated in a passive twoterminal network; Fig. 3 is a graph of the equation representing the real part of the impedance Z of the network of Fig. 2; Fig. 4 represents a component network termed an L-structure; Fig. 5 represents a four-terminal component network comprising a transformer and an L-structure; Fig. 6 represents a four-terminal component network comprising an L-structure and a trans- 4`former; Fig. 7 represents a component L-structure network; Fig. 8 represents a four-terminal component network comprising a transformer and an L-structure; Fig. 9 represents a four-terminal component network comprising an L- structure and a transformer; Fig. 10 represents a four-terminal compo-nent network comprising a transformer and a lattice structure; Fig. 11 represents a four-terminal component network comprising a lattice `structure and a transformer; Figs. 12-and 13 represent further component networks having transformers; Fig. 14 shows a II- structure component network; Fig. 15 represents a T-structure component network; Fig. 16 represents a component network whose short-circuit driving-point and transfer admittances are all equal; Fig. 17 represents a component network whose open-circuit driving-point and transfer impedancesare all equal; Fig. 18 is a general representation of a four-terminal network cornprising a plurality of component four-terminal networks in parallel; Fig. 19 is a general representation of a four-terminal network comprising a plurality of component four-terminal networks in series; Figs. 20, 21, 23, 24, 25 and 26 illustrate realizations of numerical examples, according to Method I; Fig. 22 is a diagram indicating a transformation of the location of the zeros and poles of a prescribed function.

Figs. 27, 28, 31, 32, 3'7, 38 and 40 are graphs of some of the functions employed in Method II;

Figs. 29 and 30 illustrate realizations according to Method II; Figs. 33, 34, 35 and 36 illustrate realizations of numerical examples according to Method II; Fig. 39 is a network representation of -functions realizable into a structure having only two kinds of elements, according to Method II; Fig. 41 is an equivalent network to that of Fig. 39; Figs. 42 and 43 are general representations of networks obtained by Method III; Figs. 44 and 45 are network representations of numerical examples, when applying Method III. Fig. 46 is a schematic network representation in accordance with the invention designed from admittance functions. Fig. 47 is a similar schematic network representation designed from impedance functions.

Methods of design The following sections give methods for the design of component networks as well as how to nd the total network representation of the total prescribed functions.

Suppose we have a passive four-terminal network containing n meshes, as indicated in Fig. l. Assume the applied voltages at the respective ends to be sinusoidal, of the form EN". Then, as the currents must be of the same form, i. e. IEM the well-known Kirchhofs equations for said network are:

R, L and C are the three kinds of network elements i. e., capacitance, inductance and resistance,

d P f w=21r times the frequency.

Letting D be the determinant of the zzs, i. e.

Z11212 Zin z z D 21 z2 ZnlZng Zn and Mer be the corresponding minor of the sth row and the rth column, i. e. for instance:

then, when solving for the currents in the 1st and nth mesh, being the only ones which interest us, we get, by the aid of Cramers rule:

For short-circuit at the respective ends the corresponding voltages become zero and when writing Equations (2) as and knowing that, when dealing with quantities obtained at end n, either the sign of the two end currents or of the two end voltages have to be reversed, and then we see that:

is the short-circuit driving-point admittance at end 1.

Mun

is the short-circuit driving-point admittance at end n, and

Mnl -Mln l; D -EfE1 is the short-circuit transfer admittance between the two ends.

When solving for the two voltages contained in (3) we get:

allrmfaln Il 211' 21 (111611111 06111 (Imam CV111 For open-circuit at the respective ends the corresponding currents become zero, and when writing (4) as:

and applying the sign rule, then we see that:

is the open-circuit driving-point impedance at vend 1.

...En nn` In is the open-circuit driving-point impedance at end n.

is the open-circuit transfer impedance between the two ends.

Again, solving for the two currents contained in (5), we get:

i. e. We are back to (3).

Consequently, between the ars and the rs the j y following relations hold: 557

Equations (7) and (S) reveal the interesting fact that generally the is and :s, from a purely mathematical point of view, simply are elements of two non-singular inverse square matrices the -product of which is equal to the unit matrix, often called idernfactor.

On the other hand, when solving for-E1 and I1 from Equation (3) we get:

.Bln

1 m. -En I" ln ln Comparing Equations (10) and (11) with (9), we have:

2:@1 D Ulln ln Using the quotients of the azs (say) in (12) and forming ADBC, we get:

@man ufnvan- 011112 alnaln alnaln which is equal to +1, i. e. between the general circuit parameters the relation:

AD-BC=1 (13) holds.

Further, when forming the product AD in two diierent ways, we get:

2111151111 011511 Uln ln a511118111.

Consequently, the products of the short-circuit driving-point admittance and the open-circuit driving-point impedance at the respective ends of a four-terminal network are equal.

It was stated that each one of the 01:3 and ts, because they characterize a four-terminal network having a nite number of meshes, must be prescribed as a rational fraction, as shown below:

anzgio) ho) 11(1) 1 ann h()` al mme) where the y@ :s and hm :s are rational integral polynomials in )i and the Variable )\=y+7'w refers to the complex plane; being equal to 21r times frequency.

Then, when comparing and (8), we see that the ,61s fulll the same requirement.

Each one of the ais and :s must further be real, for real )\-values, which is the same as saying that for 01:0 the response vector must be in phase with the impulse vector and both vectors must have the same direction.

Regarding the location of zeros and poles of the driving-point functions, it is known, and can readily be proved by the aid of the law of conservation of energy, that they all must lie in the left half of the complex plane, including the boundary (zthe imaginary axis).

As to the transfer function (am or in, respectively), its poles are generally the same as those of the driving-point functions.

Its zeros, however, are found to make an exception as to their location, and it can be proved that they can lie at any point of the )\plane without violating the law of conservation of energy.

From (7) and (8) it is clear that the zeros and poles of the aand -determinants must lie within the same region as those of the driving-point functions.

Then we come to the question of positiveness of the real parts of the aus and :s and the determinant of the real parts of either set of functions.

The real part of a driving-point function must necessarily be greater than or equal to Zero for 'y greater than or equal to zero, as proved by 0. Brune in his article in the Journal of Mathematics and Physics vol. X No. 3, 1931, p. 191.

As to the transfer function it should be understood that it has no physical meaning in terms of energy flow since it is equal to the quotient cf the input voltage and the output current, or vice versa. Thus, its real part is allowed to take on positive as well as negative values, though the amplitudes of these values are limited by a determinant condition to be shown below.

This determinant condition is obtained in the following manner:

Terminate the four-terminal network in a passive tWo-terrninal network, whose impedance is zzzf--y'zm, as shown in Fig. 2, and consider the whole right half of the )\plane.

Form the ratio from (10), getting:

The denominator is a sum of two squares and as in the numerator each one of the two squares Zr2 and ZX2 have the same coecient it is clear that (18) represents the equation of a parab-oloid of revolution. Because in addition the rzs and the distance between its vertex and said plane to be greater than or equal to zero, for all real values of yk 0 and all real values of w, because this restriction need to hold only for Zr-values which are greater than or equal to Zero, and it can be proved that the axis of the paraboloid never crosses the ZX-aXis, but projects through the ZT-Zz-plane only at negative or zero values O Zr.

Taking these facts into account, i. e. considering only that part of the paraboloid whose projection corresponds to positive or zero values of Zr, as indicated in Fig. 3, then it can be shown that an important necessary condition, always to be satisned if the designed network is to be passive, is that for all real wzs the determinant (rmi-fm2) must be greater than or equal to zero for real values of y greater than or equal to zero, 1. e.

(hlhn-hm) must be E0, for 'yO Further, it can be proved that when the condition (19) is satisiied for all real wzs, then the condition (21) is necessarily also satisfied, and vice versa. In addition, it can also be proved that if r1, rn and (rim-fm2) are each O, for y and all real wzs, then h1 and hn are each also O, for fyO, and if h1, hn and (mha-hing) are each for y G and all real aus, then 1'; and rn are each also El), for q/ 0.

In the foregoing we have been dealing with the whole right half of the A plane and all mentioned positiveness Iconditions refer to that region. As none of the aus or [32s have poles to the right of the boundary, however, it is clear that each one of the functions is regular in the right half of the )\plane. Consequently, in this region the functions fulll the well known Cauchys lineintegral requirements making it possible for us to limit our efforts to the boundary (the imaginary .Xis and corresponding to yz) only.

Further, all the potential functions (being the real and imaginary part of each a and as well as the determinant of these parts fulll in the right half oi the )\-plane the well known Poissons line integral requirements which means that the minimum-(as well as the maximum)- value of these functions must fall on the boundary, thus making it necessary to investigate them there only.

It has to be remembered, how-ever, that poles of any one of the ars or zs may also' fall on the boundary. When this happens the functions have to be given special attention close to such points, because such poles do not show up in the real part along the boundary of the corresponding k-function.

Thus, it is stated by Brune in his article, referred to above, that if a driving-point function has poles on the boundary, then such poles must belsimple and the residue of the function at such a pole must be a positive, real, constant.

This necessity is found when expanding the function between two concentric circles around and innitesimally close to the pole into a Laurents series and solving for the real part.

All three aor [ii-functions can readily be expanded in an analogous manner and when then solving for the real parts and applying the determinant lconditions (19) or (21), respectively, then we reach the important necessary condition, applicable for a common pole on the boundary, saying that the determinant:

(leden-km2) must be real and 0 (22) where k1, kn and km are the residues of an, ann and am or of u, [3mi and m, respectively, at the pole in question.

As to zeros on the boundary, it is known that such zeros of a driving-point function must be simple and that the differential coefficient of the function at the point in question must be a positive, real, non-zero constant. This condition is arrived at by expanding the function inside a small circle, surrounding the zero, into a Taylors series and solving for the real part.

When expanding all three ats or ,8:s in the same manner and applying (19) or (21), respectively, We then reach the necessary condition saying that the determinant:

(cult-qm2) must be real and O (23) where q1, qu and qm are the differential coefcients of the azs or zs at the zero in question.

The necessary conditions referring to zeros on the boundary are not of fundamental importance, however, because they are automatically satisfled, when the conditions summarized in the following are satisfied. Definition The process of synthesizing a finite passive four-terminal network from a prescribed matrix whose elements are the short-circuit admittances or open-circuit impedances is, in analogy with Brunes notation when dealing with a single function, called nding a network representation of the matrix.

Definition If a function fo) is realwhen )i is real, and if in addition the real part of fw is O, for vo, then such a function is after, Brune, called a positive real function. Brune also proved (see his article referred to) that such a function always can be given a network representation. Definition Let three functions property that the real part of each is 0, for Ao and if in addition:

Re fuoqX Re mm-[Re .M0012 is O, for 'yO (where Re stands for The real part of), then such a matrix is called a positive real matrix.

The fwts may stand for either the aus or the 13:3, having corresponding subscripts, and when considering the boundary, a positive real matrix is also dened by:

(I) No poles of the prescribed functions lie to the right of the imaginary axis in the A-plane.

(II) At poles on the imaginary axis residues of i110) and fana) are nite, positive, real constants.

(III) The determinant of the residues (k1, kn and 7cm) at a common pole on the imaginary axis, i. e. (Riku-km2) is a real constant and (V) Re J'uufoXRe innate-[RE finamlzo, fol' all real values of w.

Naturally, the fw :s must on the first hand fulll the requirement of being real for real Values of A.

positive real When a set of prescribed azs or zs full the requirements of being elements of a positive real matrix, in which the elements located on the principal diagonal are the two driving-point functions, then I have found that such a matrix can always be given a physical interpretation in a passive four-terminal network.

Below we shall show how to realize a positive real aor -matrix, having the specific feature that a constant ratio exists between certain or all of its elements. The network representations of such matrices will play an important rle as component networks in the total network representation of the total functions.

In the event that the prescribed functions happen initially to be elements of such a matrix, they fall under the items below directly.

(a) Let us first consider the case when the ratio between au and am (correspondingly, the ratio between nn and m) is a finite, real constant (positive or negative).

From (12) it is clear that this ratio is simply the general circuit parameter D, and when considering the ars (say) we see that in this case the prescribed functions are:

Suppose the prescribed functions fulfil the requirements of being elements of a positive rea matrix corresponding to a matrix whose elements are the general circuit parameters ABCD. This latter matrix can then be split up as:

1 B 1 AB 0 DA BD DA E E 0 X C (24) CD 0 D E 1 CD 1 0 D All component matrices of (24) we recognize immediately. The products evidently consist of a transformer matrix and a matrix specifying the L-structure shown in Fig. 4, and vice versa, for the two alternatives.

Cri"

As to positiveness of the real parts, we know that:

rlO, for 'yO 'Adding a prime to the azs which correspond to the L-structure, they are by the aid of (12) and (24) found to be:

am BD-Ezrm Consequently, We see that Ream' and Reamr are each 0, for 'y The determinant becomes:

I1 1'12 11 9 E =(Drnr1) which also is O, for y 2D, because each part of the determinant of (25) is 0, for 'y 0.

Thus, the L-structure is passive and we get the passive combination shown in Fig. 5. The

admittances of this structure (the Y:s) expressed in the general circuit parameters are simply:

VPE

Alternative II Letting the azs, referring to the L-structure, in this case be W18 we get:

(28) DZA Olrm.` B-= 2yf/m.

Thus, Reali and Ream are each 20, for

The determinant which in this case becomes:

is also O, for fy0, due to (25).

Thus, we get the passive structure shown in Fig. 6, and the Y:s are simply:

Y2=CD The reality clause is fulfilled, and due to the above two alternatives we can state:

If a non-singular positive real aor -matrix is such that the general circuit parameter D, obtained from this matrix, is a finite, real (positive or negative; if negative we only transpose one pair of terminals or wind the transformer as to take care of the negative sign) constant, then this particular matrix satisfied both necessary and sufcient conditions for having a network representation in either one of the structures shown in Fig. or 6.

Should, however, this parameter be unity, then the network representation of the corresponding matrix is shown in Fig. 4.

(b) Let us now consider the case when the ratio between am and in (correspondingly, the ratio between n and in) is a nite, real constant (positive or negative).

From (12) we see that this ratiosimply is the general circuitparameter A, and thus in this case the prescribed functions are:

aan c1h 0Lun and having the real parts:

and t" r1, ru

As these functions are elements of a positive real matrix we know that:

The corresponding matrix having as elements the general circuit parameters we in this case split up as:

AB Ao 1 1 AB Ao X X (31) CD o-1 CA DA 9 DA o-1 A A A Thus, Ream and Realm' are each O, for 'y 0, and th?. determinant:

Consequently, we get the passive combination shown in Fig. 8, and the Y:s are simply:

is also 0, for

Thus, Ream" and Ream" are each O, for yo, and the determinant:

is also O, for 'y 0, due to (30).

Consequently, we get the passive combination shown in Fig. 9, and the Yzs are simply:

The reality clause is fuliilled and thus We can state:

If a non-singular positive real aor -matrix is such that the general circuit parameter A, to be obtained from this matrix, is a nite, real The product matrices evidently specify a transformer and a symmetrical lattice structure, and vice versa.

The given aus (say) are in this case:

i. e. their real parts are We know that:

tlO, for fyO rlo, for fyO r12-r12 0, for v Alternative I Adding a prime to the ons corresponding to the lattice structure matrix, then these are found to be:

Thus, we see that Rean'=Rean is 20, for yo. The determinant, which in this case becomes:

A2 A A A 2 2 :E B11-fin is, due to (36), evidently also, for y 0.

Thus, we get as network-representation the passive combination shown in Fig. l0 and the Y:s when expressed in the general circuit parameters, are simply:

l Let the azs corresponding to the lattice structure matrix in this case be a":s, and we get:

Thus Re11"=nn 0, for 'y 0. The determinant, which here becomes:

is, due to (36), also O, for fy O.

Consequently, we get the passive combination shown in Fig. 1l and the Yzs expressed in the general circuit parameters, are:

l D YgDn/a) 1 D Een The reality clause is fulfilled, and thus we can state:

If a non-singular positive rea aor -matrix is such that the ratio between the two elements which are located on its principal diagonal, corresponding to the ratio between the general circuit parameters A and D, is a finite, real constant, then this particular matrix satisfies both necessary and suicient conditions for having a network representation in either one of the structures shown in Fig. 10 or l1. When said ratio is unity the lattice-structure alone is a network representation of the matrix considered.

(d)` Finally, we take up the case when a positive real matrix is such that a constant ratio exists between any two of the three functions. i

This is the case we continuously meet when applying my general methods of realization, because when removing sets of poles from the boundary, by the aid of a partial fraction expansion, any two members of the separate component families form always a constant ratio.

The simplest structure to use for this case is either one of the structures shown in Fig. 5, 6, 8 or 9.

If, in addition, the matrix is singular the above four structures, when considering the ons (in which case the general circuit parameter C is zero), reduce to either one of the structures shown in Fig. l2, and when considering the zs (i. e. the general circuit parameter B is zero), they reduce to either one of the structures shown in Fig. 13.

It is evident that because functions under this iteml also fall under items (a), (b) and (c) any one of the structures arrived at under these items can be used as a network representation. Further, item (c) evidently falls in between items (a) and (b).

Considering Alternative II only and letting the transformer ratio as in Fig. 6 gradually become Azl as in Fig. 9, then it can be found that for every transformer ratio, 11:1, between the above mentioned limits we get simple passive structures, while for 1L:S lying outside the range to A generally two transformers in chain connection are coming into play and when merged together the ratio is again found to lie in between the above values. Thus, it is evident that the structures of Figs. 6 and 9 form the limits for an infinite number of equivalent networks, wherefore we can state:

If a non-singular positive real aor Iisi-matrix is such that any two of the three functions bear to each other finite, real, constant ratios, then this matrix satisfies both necessary and sufcient conditions for having a network representation in any one of an innite number of equivalent structures, with those shown in Figs. 5 and 8 or 6 and 9, repectively, as limiting cases.

At least two more structures are of great importance for realization of certain positive real matrices or component families whose members are elements of such a matrix. The structures referred to are the II- and the T-(Z) structures.

(e) IThe II-structure, shown in Fig. 14, can be said to belong to the ze because when expressing the Yzs in the ons it becomes immediately clear vfor which range of functions this structure can be used as a network representation, while when expressed in the zs we get equations; which we cannot look through, and vice versa as to the T-structure.

The ons of the l'I-structure in Fig. 14 are:

au: Yil Ya am: Y2+ Ys (42) c111|: Ya

Y1: du ain: (f1 Tin) 'i-. (X1 Xin) YZIann-aln: (rn-rin) "l'j(Xn-Xln) Ya Olin 11 l-.Xin

Thus, we see that if a non-singular positive rea iL-matrix is such that the differences (ail-ain) and (ami-ain) are positive real functions and if in addition am is a positive real function itself, then this particular matrix satisnes both necessary and suincient conditions for having a network representation in a H-structure.

The T-structure, shown in Fig. 15, covers an exactly analogous range when considering the 13:3. The Zzs (which al1 have to be positive real functions) of this structure are readily found to be as follows:

Thus, if a non-singular positive real ematrix is such that the diferences (ii-m) and (fm-m) are positive real functions and if in addition [311i is a positive real function itself, then this particular matrix satisfies both necessary and suiicient conditions for having a network representation in a T-(Z) structure.

(y) Note, the 11- and T-structures as well as the two L-structures are composed ofthe two structures shown in Figs. 16 and 17. The former belongs to the ats (it can not even be defined by the aid of the 13:8) and is dened by the general circuit parameter matrix:

while the latter belongs to the rs in an analogous strict manner and is dened by the general circuit parameter matrix:

nents, and while each component family may sep-` arately be realized into av known structure we need to know in addition how to connect these structures into a unit forming a vtotal network representation of the total functions.

When considering a sum of iii-families the diagram of connections would essentially be as shown in Fig. 18, with the separate component networks connected in parallel, since they are realizations of admittance functions, and when considering a sum of -families the diagram is essentially given by Fig. 19, with the separate component networks connected in series, since they are realizations of impedance functions.

The transformers shown at one end are included in order to force the entering and leaving currents of each component structure to be equal. For a symmetrical lattice component structure usually its transformer may be omitted, however.

The insertion of a 1:1 ratio transformer may also be omitted when being part of an unsym inetrical component structure if the entering and leaving currents in the component structure in question are equal without the transformer.

On the other hand, where the transformer ratio is different from 1:1 the transformer cannot be omitted.

Further, it cannot be omitted if therewith some part of the network representation would be snorted out, as illustrated in some of the examples in the following.

Where transformers are used, it is obvious that the transformer may be placed at either end of the component structure of which it is a part.

I General methods of realization of any positive real aor ,i3-matrix having such properties, that it can characterize a passive ,foar-terminal network having two kinds of elements only (a) We now have material enough for a complete solution of the above specific cases andV start with the one when the prescribed matrix a function or its reciprocal can always be ex-` panded into partial fractions and each term can be given a physical interpretation, whereupon these are joined together in a proper way. In our case more conditions must be satisfied. but the reality clause and the residue conditions embodied in the positive real matrix definition are found to be both necessary and sucient.

The elements of a positive real aor -matrix falling under this item have each the genera form:

The denominator of the second rational fraction must then be factored. Since we know that every pole necessarily lies on the imaginary axis and that such poles must appear in complex pairs, the factors must be A, (Mi-iwi), ()\1'w1), (A4-jez), ()\-9'w2) etc. The fraction to the right can then be still further expanded, as will be evident to those skilled in the art. To save space we can now greatly simplify the resultant expansion by adopting an arbitrary set of emcients, as indicated below, whereby the numerators of the fractions appear in the form generally obtained in partial fraction expansions. Thus (46) are found to be nothing but residues of fw at respective poles (located on the imaginary axis at w, the origin, and at points corresponding to im, i'waim, etc.) and because these residues,v due to our necessary residue conditions, have to be real we see that all lczs must be zero, wherefore (47) goes over into:

Il, i

where 7c, ko, k1, k2, etc. are the residues' of ,fw at respective poles along the boundary.

The coeiiicients in (48) may be evaluated from the completed expansion of which the rst step is shown in (45a). Thus from this equation it is already evident that However, these coeiicients, being residues, may

also be found directly from (45). Thus since ko is the residue of fm at the pole located at the origin,

As fw may stand for any one of the three aus or zs and as each member of the same family generally has the same denominator, a similar expansion as (48) is possible for each one ofthemiand we get, when considering the aus, say:

Here, as well `as in the following when meeting similar expansions of the three aus orzs the dilferent columns, i. e. k f

ko kol ko/l. 4 i

Vflw12 Nfl-w12 X24-w12 are called component families. 'The members of such a component family form elements of a component matrix and are called component aus or component rs respectively, while the network representation of each component family is called a component network.

We vnow immediately see that (49) has `a network representation in aV structure as schematically shown in Fig. 18, andif (49) had been the expansions of the three zs then `the network representationv would have been a structure as schematically shown in Fig. 19, although in la, km, kfx

both cases any one of :the four-terminal com--V ponent networks shown in said figures may happen to be-a two-terminal network.

The structure in question, whetherin accordance with Figs. 18 or 19, can readily be built because any two members of each component family bear to each other a finite, real, constant ratio, i. e. each component family has a network representationinany one of an infinite number of equivalent structures.

Further, these component structures are all passive because the conditions'of a positive real matrix, derived from the' consideration of such structures, are satised. (Note, lthe only items, under the definition of such a matrix, to consider are (i), (ii), and y(iii), i. e. the location of poles and the'residue conditions.) Further, naturally the reality clause is fullled.

Thus, we can state:

If a prescribed .positive real` aor -matrix is such that all zeros and poles of its elements lie on the imaginary axis in the l-plane, then this particular matrix satisfies both. necessary and sufficient conditions for having a network representation in a purely reactive structure in a combination as shown in Figs. 18 or 19, respectively.

For an illustration, consider the elements o'f a The above functions full all requirements of 

